Bigalois extensions and the graph isomorphism game

Abstract

We study the graph isomorphism game that arises in quantum information theory from the perspective of bigalois extensions of compact quantum groups. We show that every algebraic quantum isomorphism between a pair of (quantum) graphs $X$ and $Y$ arises as a quotient of a certain measured bigalois extension for the quantum automorphism groups $G_X$ and $G_Y$ of the graphs $X$ and $Y$. In particular, this implies that the quantum groups $G_X$ and $G_Y$ are monoidally equivalent. We also establish a converse to this result, which says that every compact quantum group $G$ monoidally equivalent to $G_X$ is of the form $G_Y$ for a suitably chosen quantum graph $Y$ that is quantum isomorphic to $X$. As an application of these results, we deduce that the $\ast$-algebraic, C${}^{\ast }$-algebraic, and quantum commuting (qc) notions of a quantum isomorphism between classical graphs $X$ and $Y$ all coincide. Using the notion of equivalence for non-local games, we deduce the same result for other synchronous non-local games, including the synBCS game and certain related graph homomorphism games.

33 pages

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Cited by in corpus (20)

• The quantum-to-classical graph homomorphism game
• Quantum graphs: different perspectives, homomorphisms and quantum automorphisms
• Some examples of quantum graphs
• Classification of Quantum Graphs on $M_2$ and their Quantum Automorphism Groups
• Quantum no-signalling correlations and non-local games
• Random quantum graphs
• Synchronous linear constraint system games
• Unitary transformations of fibre functors
• Quantum automorphism groups of connected locally finite graphs and quantizations of discrete groups
• Quantum Semigroups from Synchronous Games
• Quantum teleportation in the commuting operator framework
• The Frucht property in the quantum group setting
• Entanglement-symmetries of covariant channels
• Quantum graphs, subfactors and tensor categories I
• Non-local games and quantum symmetries of quantum metric spaces
• Asymmetric graphs with quantum symmetry
• Duals of finite quantum permutation groups
• Algebraic connectedness and bipartiteness of quantum graphs
• Complete Classification of Directed Quantum Graphs on M2
• Quantum partial automorphisms of finite graphs